3U XI ELEMENTARY SPECTRAL THEORY

Show that (w^(eh) \ ek) ^ 0 for every pair h> k> and WN =^ 0 by assumption, and use the
fact that WW>N = p(w)wN.)

(d) Suppose that (u(ek) \ ek) > 0 for every pair of indices /z, k (this, by (b), implies
p(u) > 0), and in addition suppose that p(u) is a pole of i?c. Show that p(u) is then a
simple pole of t?c. (Observe that (djdQ — v^ = vf and prove that, if one had N > 1, then
one would have w§ = 0; this would imply (wN(ek) \ ek) = 0 for every &•; using the relation
uwx — p(ti)wN> observe that if (wN(ek) \ eh) = 0 for one index h, then WN(^) == 0.) Prove
that there exists an eigenvector z=£ £„*„ of u corresponding to />(«) and such that

n

£w > 0 for every n. Next show that if y = £ ??„ en is an eigenvector of the adjpint u*

n

corresponding to p(u) (cf. Section 11.5), one has 7]n ^ 0 for every 77, or rjn ^ 0 for every n
(otherwise, one would get the contradictory inequality £ £n I fyi I <2 £« | fyil, using a

n u

majoration of each rjn derived from p(u)r]n =X yk(u(ek) \ en)). Conclude finally that all

eigenvectors of u corresponding to p(u) are scalar multiples of z (exchange u and u*).
(" Theorem of Frobenius-Perron.")

2, COMPACT OPERATORS

Let E, F be two normed (real or complex) spaces; we say that a linear
mapping u of E into F is compact if, for any bounded subset B of E, u(E)
is relatively compact in F. An equivalent condition is that for any bounded
sequence (xn) in E, there is a subsequence (xn]) such that the sequence (u(xnj)
converges in F. As a relatively compact set is bounded in F (3.17.1), it follows
from (5.5.1) that a compact mapping is continuous.

Examples

(11.2.1) If E ©F is finite dimensional, every continuous linear mapping
of E into F is compact (by (5.5.1), (3.17.6), (3.20.16), and (3.17.9)).

(11.2.2) If E is an infinite dimensional normed space, the identity operator
in E is not compact, by F. Riesz's theorem (5.9.4).

(11.2.3) Let I = [a, b] be a compact interval in R, E = #C(I) the Banach
space of continuous complex-valued functions in I (Section 7.2), (s, t) -> K(s, t)
a continuous complex-valued function in I x I. For any function /e E, the

mapping t-+( K(s,t)f(s)ds is continuous in I by (8.11.1); denote this

*/ **nd an example of an